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Calculus for Team-Based Inquiry Learning PREVIEW Edition

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Section 3.6 Concavity and Inflection (AD6)

Subsection 3.6.1 Activities

Observation 3.6.1.

In addition to asking whether a function is increasing or decreasing, it is also natural to inquire how a function is increasing or decreasing. Activity 3.6.2 describes three basic behaviors that an increasing function can demonstrate on an interval, as pictured in Figure 56
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Activity 3.6.2.

Sketch a sequence of tangent lines at various points to each of the following curves in Figure 56.
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Diagram Exploration Keyboard Controls
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Figure 56. Three increasing functions
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(a)
Look at the curve pictured on the left of Figure 56. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
C: The slopes of the tangent lines increase as you move from left to right.
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(b)
Look at the curve pictured in the middle of Figure 56. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
B: The slopes of the tangent lines remain constant as you move from left to right.
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(c)
Look at the curve pictured on the right of Figure 56. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
A: The slopes of the tangent lines decrease as you move from left to right.
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Remark 3.6.3.

On the leftmost curve in Figure 56, as we move from left to right, the slopes of the tangent lines will increase. Therefore, the rate of change of the pictured function is increasing, and this explains why we say this function is increasing at an increasing rate.
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Observation 3.6.4.

We must be extra careful with our language when dealing with negative numbers. For example, it can be tempting to say that “\(-100\) is bigger than \(-2\text{.}\)” But we must remember that “greater than” describes how numbers lie on a number line: \(-100\) is less than \(-2\) becomes it comes earlier on the number line. It might be helpful to say that “\(-100\) is "more negative" than \(-2\text{.}\)”
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Activity 3.6.5.

Sketch a sequence of tangent lines at various points to each of the following curves in Figure 57.
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
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Y Stop sound output
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P Repeat sonification output
Figure 57. From left to right, three functions that are all decreasing.
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(a)
Look at the curve pictured on the left in Figure 57. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
C: The slopes of the tangent lines increase as you move from left to right.
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(b)
Look at the curve pictured in the middle in Figure 57. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
B: The slopes of the tangent lines remain constant as you move from left to right.
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(c)
Look at the curve pictured on the right in Figure 57. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines remain constant as you move from left to right.
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  3. The slopes of the tangent lines increase as you move from left to right.
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Answer.
A: The slopes of the tangent lines decrease as you move from left to right.
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Remark 3.6.6.

Recall the terminology of concavity: when a curve bends upward, we say its shape is concave up. When a curve bends downwards, we say its shape is concave down.
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Activity 3.6.7.

Look at the curves in Figure 58. Which curve is concave up? Which one is concave down? Why? Try to explain using the graph!
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
Figure 58. Two concavities, which is which?
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Answer.
The curve on the left is concave up because it bends up from its tangent lines, while the curve on the right is concave down because it bends down from its tangent lines.
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Definition 3.6.8.

Let \(f\) be a differentiable function on some interval \((a,b)\text{.}\) Then \(f\) is concave up on \((a,b)\) if and only if \(f'\) is increasing on \((a,b)\text{;}\) \(f\) is concave down on \((a,b)\) if and only if \(f'\) is decreasing on \((a,b)\text{.}\)
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Activity 3.6.9.

Look at how the slopes of the tangent lines change from left to right for each of the two graphs in Figure 58
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(a)
Look at the curve pictured on the left in Figure 58. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines increase as you move from left to right.
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  3. The slopes of the tangent lines go from increasing to decreasing as you move from right to left.
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  4. The slopes of the tangent lines go from decreasing to increasing as you move from right to left.
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Answer.
B: The slopes of the tangent lines increase as you move from left to right.
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(b)
Which of the following statements is true about the function on the left in Figure 58?
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  1. \(f'(x) > 0 \) on the entire interval shown.
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  2. \(f'(x) < 0 \) on the entire interval shown.
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  3. \(f''(x) > 0 \) on the entire interval shown.
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  4. \(f''(x) < 0 \) on the entire interval shown.
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Answer.
C: \(f''(x) > 0 \) on the entire interval shown.
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(c)
Look at the curve pictured on the right in Figure 58. How would you describe the slopes of the tangent lines as you move from left to right?
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  1. The slopes of the tangent lines decrease as you move from left to right.
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  2. The slopes of the tangent lines increase as you move from left to right.
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  3. The slopes of the tangent lines go from increasing to decreasing as you move from right to left.
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  4. The slopes of the tangent lines go from decreasing to increasing as you move from right to left.
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Answer.
A: The slopes of the tangent lines decrease as you move from left to right.
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(d)
Which of the following statements is true about the function on the right in Figure 58?
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  1. \(f'(x) > 0 \) on the entire interval shown.
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  2. \(f'(x) < 0 \) on the entire interval shown.
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  3. \(f''(x) > 0 \) on the entire interval shown.
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  4. \(f''(x) < 0 \) on the entire interval shown.
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Answer.
D: \(f''(x) < 0 \) on the entire interval shown.
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Observation 3.6.11.

In the previous section, we saw in Activity 3.5.8 how to use critical points of the function and the sign of the first derivative to identify intervals of increase/decrease of a function. The next activity Activity 3.6.12 uses the critical points of a first derivative function and the sign of its second derivative (accordingly to Theorem 3.6.10) to identify where the original function is concave up/down.
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Activity 3.6.12.

Let \(f(x)=x^4-54x^2\text{.}\)
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(a)
Find all the zeros of \(f''(x)\text{.}\)
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Answer.
\(x = \pm 3\)
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(b)
What intervals have been created by subdividing the number line at zeros of \(f''(x)\text{?}\)
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Answer.
\((-\infty,-3)\text{,}\) \((-3,3)\text{,}\) and \((3,\infty)\text{.}\)
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(c)
Pick an \(x\)-value that lies in each interval. Determine whether \(f''(x)\) is positive or negative at each point.
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Answer.
\(f''(x)\) is positive on \((-\infty,-3)\cup (3,\infty)\) and negative on \((-3,3)\text{.}\)
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(d)
On which intervals is \(f'(x)\) increasing? On which intervals is \(f'(x)\) decreasing?
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Answer.
\(f'(x)\) is increasing on \((-\infty,-3)\cup(3,\infty)\) and decreasing on \((-3,3)\text{.}\)
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(e)
List all the intervals where \(f(x)\) is concave up and all the intervals where \(f(x)\) is concave down.
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Answer.
\(f(x)\) is concave up on \((-\infty,-3)\cup (3,\infty)\) and concave down on \((-3,3)\text{.}\)
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Definition 3.6.13.

If \(x=c\) is a point where \(f''(x)\) changes sign, then the concavity of graph of \(f(x)\) changes at this point and we call \(x=c\) an inflection point of \(f(x)\text{.}\)
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Activity 3.6.14.

Use the results from Activity 3.6.12 to identify all of the inflection points of \(f(x)=x^4-54x^2\text{.}\)
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Answer.
The inflection points occur at \(x = \pm 3\text{.}\)
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Activity 3.6.15.

For each of the following functions, describe the open intervals where it is concave up or concave down, and any inflection points.
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(a)
\(f(x)=-\frac{1}{4} \, x^{5} - \frac{5}{2} \, x^{4} - \frac{15}{2} \, x^{3}\)
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Answer.
\(f(x)\) is concave up on \((-\infty,0)\) and concave down on \((0,\infty)\text{.}\) It has an inflection point at \(x = 0\text{.}\)
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(b)
\(f(x)=\frac{3}{20} \, x^{5} + x^{4} - \frac{5}{2} \, x^{3}\)
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Answer.
\(f(x)\) is concave down on \((-\infty,-5)\cup (0,1)\) and concave up on \((-5,0)\cup (1,\infty)\text{.}\) It has inflection points at \(x = -5\text{,}\) \(x = 0\text{,}\) and \(x = 1\text{.}\)
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(c)
\(g(x) = x - \cos\left(\dfrac{\pi}{2}x\right)\) on \((0,2\pi)\)
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Answer.
\(g(x)\) is concave up on \((0,1)\cup (3,5)\) and concave down on \((1,3)\cup (5,2\pi)\text{.}\) It has inflection points at \(x = 1\text{,}\) \(x = 3\text{,}\) and \(x = 5\text{.}\)
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Activity 3.6.16.

Consider the following table. The values of the first and second derivatives of \(f(x)\) are given on the domain \([0,7]\text{.}\) The function \(f(x)\) does not suddenly change behavior between the points given, so the table gives you enough information to completely determine where \(f(x)\) is increasing, decreasing, concave up, and concave down.
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\begin{equation*} \begin{array}{c|cccccccc} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline f'(x) & 2 & 0 & -2 & 0 & 2 & 1 & 0 & -1 \\\hline f''(x) & -2 & -1 & 0 & 1 & 0 & -1 & 0 & 3 \\ \end{array} \end{equation*}
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(a)
List all the critical points of \(f(x)\) that you can find using the table above.
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Answer.
The critical points are at \(x = 1\text{,}\) \(x = 3\text{,}\) and \(x = 6\text{.}\)
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(b)
Use the First Derivative Test to classify the critical numbers (decide if they are a max or min). Write full sentence stating the conclusion of the test for each critical number.
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Answer.
There are local maxima at \(x = 1\) and \(x = 6\) and a local minimum at \(x = 3\text{.}\)
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(c)
On which interval(s) is \(f(x)\) increasing? On which interval(s) is \(f(x)\) decreasing? List all the critical points of \(f(x)\) that you can find using the table above.
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Answer.
\(f(x)\) is increasing on \((0,1)\cup (3,6)\) and decreasing on \((1,3)\cup (6,7)\text{.}\)
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(d)
There is one critical number for which the Second Derivative Test is inconclusive. Which one? You can still determine if it is a max or min using the First Derivative Test!
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Answer.
The Second Derivative Test is inconclusive at \(x = 6\text{.}\)
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(e)
List all the critical points of \(f'(x)\) that you can find using the table above.
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Answer.
\(f'(x)\) has critical points at \(x=2\text{,}\) \(x = 4\text{,}\) and \(x = 6\text{.}\)
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(f)
On which intervals is \(f(x)\) concave up? On which intervals is \(f(x)\) concave down?
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Answer.
\(f(x)\) is concave up on \((2,4)\cup (6,7)\) and concave down on \((0,2)\cup (4,6)\text{.}\)
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(g)
List all the inflection points of \(f(x)\) that you can find using the table above.
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Answer.
The inflection points are at \(x = 2\text{,}\) \(x=4\text{,}\) and \(x=6\text{.}\)
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Subsection 3.6.2 Videos

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Subsection 3.6.3 Exercises

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